Abstract
In this paper, we study the k-tree partition problem which is a partition of the set of edges of a graph into k edge-disjoint trees. This problem occurs at several places with applications e.g. in network reliability and graph theory. In graph drawing there is the still unbeaten (n − 2) ×(n − 2) area planar straight line drawing of maximal planar graphs using Schnyder’s realizers [15], which are a 3-tree partition of the inner edges. Maximal planar bipartite graphs have a 2-tree partition, as shown by Ringel [14]. Here we give a different proof of this result with a linear time algorithm. The algorithm makes use of a new ordering which is of interest of its own. Then we establish the NP-hardness of the k-tree partition problem for general graphs and k ≥ 2. This parallels NP-hard partition problems for the vertices [3], but it contrasts the efficient computation of partitions into forests (also known as arboricity) by matroid techniques [7].
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Biedl, T., Brandenburg, F.J. (2007). Partitions of Graphs into Trees. In: Kaufmann, M., Wagner, D. (eds) Graph Drawing. GD 2006. Lecture Notes in Computer Science, vol 4372. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-70904-6_41
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DOI: https://doi.org/10.1007/978-3-540-70904-6_41
Publisher Name: Springer, Berlin, Heidelberg
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